Definition: Any whole number that can be divided by 1 and itself ONLY is called a prime number.

Example 1

2 is a prime number, why?

2 can be divided by 1     2 ÷ 1 = 2 with remainder equal to zero

2 can be divided by 2 (itself)     2 ÷ 2 = 1 with remainder equal to zero

Try to find another whole number that divides 2 with remainder zero. There is not.

Example 2

7 is a prime number, why?

7 can be divided by 1     7 ÷ 1 = 7 with remainder equal to zero

7 can be divided by 7 (itself)     7 ÷ 7 = 1 with remainder equal to zero

Try to find another whole number that divides 7 with remainder zero. There is not.

The first 10 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

You can generate more prime numbers and test them.

Composite Numbers

Definition: Any whole number that can be divided (with remainder equal to zero) by another whole number other than 1 and itself is called composite number.

4 is a composite number, why?

4 can be divided by 1, itself and 2.

6 is a composite number, why?

6 can be divided by 1, itself, 2 and 3.

12 is a composite number : it can be divided by 1, itself, 2, 3, 4 and 6.

30 is a composite number: it can be divided by 1, itself, 2, 3, 5, 6, 10 and 15.

Factorization

From division to multiplication to factoring.

Division and multiplication are related operations.

The division 6 ÷ 3 = 2 may be written as a multiplication: 6 = 2 × 3

Examples

1) 6 = 1 × 6 ; 6 = 2 × 3     1, 2, 3 and 6 are called factors of 6.

2) 12 = 12 ×1 = 3 ×4 = 6 ×2     1, 2, 3, 4, 6 and 12 are called factors of 12.

3) 20 = 1 × 20 = 2 ×10 = 2 × 2 × 5 = 4 × 5     1, 2, 3, 4, 5, 10 and 20 are called factors of 20.

Prime factorization

Examples

1) 6 = 2 × 3     the factors 2 and 3 are prime numbers.

2) 12 = 2 × 2 × 3     the factors 2 and 3 are prime numbers.

3) 20 = 2 × 2 × 5     the factors 2 and 5 are prime numbers.

Example 1

Write the prime factorization of 12

1) See if the first prime number 2 is a factor of the given number 12

12 ÷ 2 = 6 with remainder = 0        2 is a factor of 12        12 = 2 × 6

2) See if the first prime number 2 is a factor of 6

6 ÷ 2 = 3 with remainder = 0        2 is a factor of 6        6 = 2 × 3        Hence 12 = 2 × 6 = 2 × 2 × 3

12 = 2 × 2 × 3 is completely factored using only prime numbers 2 and 3

Example 2

Write the prime factorization of 21

1) See if the first prime number 2 is a factor of the given number 21

21 ÷ 2 = 10 but remainder = 1 so 2 is not a factor of 21

2) Is the next prime number 3 a factor of 21?

21 ÷ 3 = 7 with remainder 0        3 is a factor of 21        21 = 3 × 7

3 and 7 are prime numbers and therefore 21 = 3 × 7 is completely factored using only prime numbers 3 and 7